As computers have become increasingly commonplace in homes and businesses throughout the world, and such computers have become increasingly interconnected via networks (such as the Internet), security and authentication concerns have become increasingly important. One manner in which these concerns have been addressed is the use of a cryptographic technique involving a key-based cipher. Using a key-based cipher, sequences of intelligible data (typically referred to as plaintext) that collectively form a message are mathematically transformed, through an enciphering process, into seemingly unintelligible data (typically referred to as ciphertext). The enciphering can be reversed, allowing recipients of the ciphertext with the appropriate key to transform the ciphertext back to plaintext, while making it very difficult, if not nearly impossible, for those without the appropriate key from recovering the plaintext.
Public-key cryptographic techniques are one type of key-based cipher. In public-key cryptography, each communicating party has a public/private key pair. The public key of each pair is made publicly available (or at least available to others who are intended to send encrypted communications), but the private key is kept secret. In order to communicate a plaintext message using encryption to a receiving party, an originating party encrypts the plaintext message into a ciphertext message using the public key of the receiving party and communicates the ciphertext message to the receiving party. Upon receipt of the ciphertext message, the receiving party decrypts the message using its secret private key, and thereby recovers the original plaintext message.
The RSA (Rivest-Shamir-Adleman) method is one well-known example of public/private key cryptology. To implement RSA, one generates two large prime numbers p and q and multiplies them together to get a large composite number N, which is made public. If the primes are properly chosen and large enough, it will be practically impossible (i.e., computationally infeasible) for someone who does not know p and q to determine them from just knowing N. However, in order to be secure, the size of N typically needs to be more than 1,000 bits. In some situations, though, such a large size makes the numbers too long to be practically useful.
One such situation is found in authentication, which can be required anywhere a party or a machine must prove that it is authorized to access or use a product or service. An example of such a situation is in a product ID system for a software program(s), where a user must enter a product ID sequence stamped on the outside of the properly licensed software package as proof that the software has been properly paid for. If the product ID sequence is too long, then it will be cumbersome and user unfriendly.
Additionally, not only do software manufacturers lose revenue from unauthorized copies of their products, but software manufacturers also frequently provide customer support, of one form or another, for their products. In an effort to limit such support to their licensees, customer support staffs often require a user to first provide the product ID associated with his or her copy of the product for which support is sought as a condition for receiving support. Many current methods of generating product IDs, however, have been easily discerned by unauthorized users, allowing product IDs to be generated by unauthorized users.
Given the apparent ease with which unauthorized users can obtain valid indicia, software manufacturers are experiencing considerable difficulty in discriminating between licensees and such unauthorized users in order to provide support to the former while denying it to the latter. As a result, manufacturers often unwittingly provide support to unauthorized users, thus incurring additional and unnecessary support costs. If the number of unauthorized users of a given software product is sufficiently large, then these excess costs associated with that product can be quite significant. Therefore, a need exists in the art for a technique that permits a software manufacturer to appreciably reduce the incidence of unauthorized copying of its software product, but which is not based on user entry of impractically long data sequences.
The invention addresses these problems and provides a cryptosystem based on a Jacobian of a curve.